C^*-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Let V = B(H) or S(H), where B(H) is the algebra of a bounded linear operator acting on the Hilbert space H, and S(H) is the set of selfadjoint operators in B(H). Denote the numerical range of A ∈ B(H) by W(A) = {(Ax,x): x ∈ H, (x,x) = 1}. It is shown that a surjective map φ: V → V satisfies W(AB + BA) = W(φ(A)φ(B) + φ(B)φ(A)) for all A, B ∈ V if and only if there is a unitary operator U ∈ B(H) such that φ has the form X ±U^*XU or X ±U^*X^t U, where X^t is the transpose of X with respect to a fixed orthonormal basis. In other words, the map φ or-φ is a C^*-isomorphism on B(H) and a Jordan isomorphism on S(H). Moreover, if H has finite dimension, then the surjective assumption on φ can be removed.